Improved Measurement Cost Scaling in the Nonorthogonal Quantum Eigensolver

K. Birgitta Whaley; Mingyu Kang

REVIEW 1 major objections 4 minor 47 references

Improved Measurement Cost Scaling in the Nonorthogonal Quantum Eigensolver

T0 review · 1 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Overlap thresholding reduces the quantum eigensolver measurement cost from cubic to linear in the number of reference states.

desk verdict A useful condition-number bound and a credible O(M) shot-count improvement, but the thresholding proof uses the exact overlap matrix while the protocol sees noise, leaving a gap that should be fixed. read the letter →

arxiv 2608.12830 v1 pith:U4XFZ4UU submitted 2026-08-13 quant-ph

Mingyu Kang , K. Birgitta Whaley This is my paper · ORCID

classification quant-ph MSC 65F1515A4281P68 PACS 03.67.Lx

keywords nonorthogonalquantumeigensolversubspacediagonalizationmeasurementcostoverlapthresholdingconditionnumberfinite-shotperturbationtheorygeneralizedeigenvalueproblemhydrogenchainsandrings

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to remove a measurement bottleneck in the nonorthogonal quantum eigensolver (NOQE), a quantum-chemistry algorithm that builds a small Hamiltonian in a subspace of dressed nonorthogonal reference states and diagonalizes it classically after estimating the matrix elements with a quantum computer. The known bottleneck is that finite-shot noise in those matrix elements is magnified by an overlap matrix that is nearly singular, so the number of circuit repetitions per matrix element can grow steeply with the number of reference states $M$. The paper's central claim is that thresholding away small-overlap directions makes the eigenvalue error depend on the condition number of the retained overlap matrix instead of on the subspace dimension, and that under a scalable thresholding scheme the per-element shot count sufficient for a target eigenangle error $\mathcal{E}$ is $\mathcal{O}(M/\mathcal{E}^2)$, improving the previously known $\mathcal{O}(M^3)$ bound. Numerical experiments on hydrogen chains and rings show the retained condition number staying controlled and the thresholding bias not growing with system size, indicating that the linear bound is realized in practice and may even be beaten.

What carries the argument

The load-bearing object is the Mathias-Li eigenangle perturbation bound, which measures eigenvalue errors of a definite Hermitian pair $(H,S)$ through the generalized eigenangle $\theta_j = \tan^{-1}(1/E_j)$: the error is at most $\sin^{-1}\big(\chi \|X\|_2 / d_{\min}\big)$, where $X$ is the unit-column matrix diagonalizing $H+iS$, $\chi$ is the size of the matrix perturbation, and $d_{\min}$ is the smallest magnitude among the normalized eigenvalue pairs. The paper's key move is Lemma 1, which replaces the crude bound $\|X\|_2 \le M$ by $\|X\|_2 \le \kappa(S)$, where $\kappa(S)$ is the overlap condition number. Thresholding (Definition 1) deletes the overlap directions with eigenvalues below $\varepsilon$, and a scalable thresholding scheme (Definition 2) is one that keeps $\kappa(S_{>\varepsilon}) = \mathcal{O}(1)$ while introducing only bounded bias in the target eigenvalue. Corollary 2 then combines these ingredients to give the $\mathcal{O}(M)$ shot-count bound. The thresholding step is what converts a nearly singular overlap matrix into a well-conditioned retained one, and that conversion is what removes the cubic dimension factor from the cost.

What would settle it

Run the thresholding rule $\varepsilon_N = cM(N)$ on a strongly correlated molecular family beyond the tested hydrogen chains and rings, scanning $c$: if for every $c$ the retained condition number $\kappa(S_{>\varepsilon_N})$ grows with $N$ while the thresholding bias stays below chemical accuracy, then the $\mathcal{O}(M)$ bound does not hold for that family. A sharper check is to build the thresholding projector from the noisy measured overlap matrix rather than the exact one and test whether the retained eigenvalues or the final energy error diverge as $N$ grows.

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Extended reading notes

Core claim

The discovery is a perturbation-theoretic reduction of the measurement cost of thresholded NOQE. For a definite Hermitian pair $(H,S)$, the paper proves that the sensitivity of generalized eigenvalues to finite-shot noise is controlled by $\kappa(S_{>\varepsilon})$, the condition number of the overlap matrix after removing eigenvectors whose overlap eigenvalues fall below a threshold $\varepsilon$, rather than by the worst-case dimension factor $M$. With a scalable thresholding scheme, defined as one where $\kappa(S_{>\varepsilon}) = \mathcal{O}(1)$ and the thresholding bias on the target eigenvalue stays $\mathcal{O}(1)$, the sufficient number of shots per matrix element to reach eigenangle accuracy $\mathcal{E}$ is $N_{\mathrm{shots}} = \mathcal{O}\big(M/(d_{\min,\varepsilon}^2 \mathcal{E}^2)\big) = \mathcal{O}(M/\mathcal{E}^2)$. This replaces the previously known $\mathcal{O}(M^3)$ upper bound. Numerical experiments on linear hydrogen chains and rings in a strongly correlated regime show that the retained condition number remains controlled as the number of reference states grows, while the thresholding bias does not visibly increase, and the ground-state energy error stays close to full configuration interaction.

Load-bearing premise

The entire linear shot-cost bound rests on the existence of a scalable thresholding scheme: a threshold that keeps the retained overlap matrix well-conditioned without shifting the target eigenvalue by more than a bounded amount at every system size, something the paper demonstrates numerically for hydrogen chains and rings but does not prove in general.

Editorial extensions

If this is right

For any NOQE instance that admits a scalable thresholding scheme, the per-matrix-element shot count sufficient to reach a fixed eigenangle accuracy grows at most linearly with the number of reference states $M$, rather than cubically.
Because the perturbation bound and thresholding machinery treat only the pair $(H,S)$, the same linear improvement carries over to any quantum subspace diagonalization method whose projected matrix elements are estimated from quantum circuits.
For an $\eta$-scalable thresholding scheme, the final energy error is bounded by the thresholding bias $\eta$ plus a contribution proportional to the eigenangle tolerance, so the two error sources combine additively rather than multiplicatively.
In the hydrogen-chain and hydrogen-ring families tested, the raw overlap condition number grows with $M$, but the thresholded condition number stays controlled, showing that the discarded small-overlap directions are not needed for ground-state energy accuracy.
The numerics further suggest that for the ground state the relevant sensitivity scale is the ground state's own eigenvalue magnitude rather than the worst-case $d_{\min,\varepsilon}$, which would make the practical cost grow even more slowly than linear; the paper identifies the stronger perturbation assumption needed to prove this.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof's thresholding projector is built from the exact overlap matrix, whereas a real protocol only has a noisy estimate; an extension that tracks how noise in the eigenvectors used for thresholding changes the retained subspace would be needed before the bound applies verbatim to hardware data.
The linear scaling counts the number of reference states $M$ as the resource, but in the tested construction $M(N) = \binom{N}{\lceil N/2\rceil}$ grows exponentially with the number of atoms, so reaping the benefit in practice requires a compact reference-subspace selection rule that avoids enumerating all spin assignments.
A natural stress test is to run the same fixed-$c$ thresholding rule $\varepsilon_N = cM(N)$ on other strongly correlated systems, such as spin defects or transition-metal clusters; if the retained condition number starts growing there while the bias stays bounded, the scalable regime is not universal.
Using quantum amplitude estimation to reduce the per-element variance from $\mathcal{O}(1/N_{\mathrm{shots}})$ to $\mathcal{O}(1/N_{\mathrm{shots}}^2)$ would likely convert the $1/\mathcal{E}^2$ factor in the bound into $1/\mathcal{E}$ while keeping the linear dependence on $M$, a direction the paper notes but does not develop.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: the paper has a real idea. Lemma 1 replaces the crude dimension bound ||X||^2 ≤ M in the Mathias-Li perturbation theory with ||X||^2 ≤ κ(S), where κ is the overlap condition number. That is one clean observation, and it is what lets the paper claim O(M) per-matrix-element shot count under thresholding, instead of the O(M^3) bound from [20,23]. I verified the proof of Lemma 1; it is short and correct. Corollary 2's algebra checks out under its stated definitions. The numerics on hydrogen chains and rings, with public code, support the claim that practical thresholding can keep the retained condition number controlled while the bias stays flat.

The soft spot is exactly the one the stress-test flags. The theory thresholds using the exact overlap matrix S: in Def. 1 and the proof of Corollary 2, the retained pair is built from V_>ε, the eigenvectors of S. A finite-shot protocol only has S+ΔS, so the thresholding eigenvectors are themselves noisy. The proof passes the perturbation through a fixed subspace, but in reality the subspace moves, and the unaccounted error involves Ṽ−V, governed by the spectral gap of S near ε (Davis–Kahan), not by χκ(S_>ε). That term is absent from the bound. So the O(M) result is not actually proved for the protocol as executed. Fig. 2's caption doesn't say which S was used for thresholding, which weakens the empirical support. This is an addressable gap, not a fatal one: either analyze the subspace perturbation under a separation assumption, or test numerically whether thresholding with the noisy S changes the scaling. I'd like to see either before the claim is stated as proven.

Two smaller points. The scalable-thresholding definition requires existence of constants c with bias O(1), and the evidence is only small-system numerics (M up to 126, per-family c). The paper is honest about that. And the bound is per matrix element, so total measurement cost is O(M^3), not O(M); the abstract does say per-element, but it's easy to misread.

The citation pattern is fine: the old bound is attributed to both [20] and [23], and the new step is clearly marked. Given the clean math, public code, and a genuinely useful bound, this deserves serious peer review. The gap between the theorem and the protocol should be closed before publication, but the paper is well worth engaging with.

Referee Report

1 major / 4 minor

Summary. The paper analyzes the finite-shot measurement cost of the nonorthogonal quantum eigensolver (NOQE). For a projected Hamiltonian-overlap pair (H,S), it uses the Mathias-Li perturbation theory for definite generalized eigenvalue problems, proves Lemma 1 that the unit-column diagonalizing matrix X satisfies ||X||^2 <= kappa(S), and then, under a formal 'scalable thresholding scheme' (Definition 2), derives in Corollary 2 that the per-matrix-element shot count sufficient for eigenangle accuracy E is O(M/(d_min^2 E^2)) = O(M/E^2), improving on the previously known O(M^3) bound. Numerical experiments on linear hydrogen chains and rings with N=2..9 and M up to 126 show controlled thresholded condition numbers and small thresholding bias, and the paper suggests the practical cost may grow even more slowly than linear.

Significance. If the central claim holds for an implementable protocol, this is a significant improvement: it replaces a cubic dependence on the number of reference states with a linear dependence for the per-element shot budget, which directly addresses a known bottleneck of quantum subspace diagonalization. The conditional theorem is mathematically coherent: Lemma 1 is a clean and correct replacement of the dimension factor by the overlap condition number, and the proof of Corollary 2 is internally consistent under the stated definitions. The manuscript also ships reproducible code and gives a concrete noise model, which are strengths. The main weakness is that the theorem is proven for thresholding with the exact overlap matrix, while any real finite-shot protocol must threshold with a noisy overlap matrix; this gap is not analyzed and is load-bearing for the claimed O(M) scaling.

major comments (1)

[Definition 2 / Numerical evidence] The O(M) improvement is conditional on the existence of a scalable thresholding scheme (Definition 2), and the manuscript does not prove that such a scheme exists for any nontrivial family; it offers numerical evidence only for hydrogen chains and rings with N=2..9, M(N) up to 126, and hand-chosen constants c=10^{-3}, 3x10^{-3}, 5x10^{-3} (Figure 2). The condition-number and bias plots therefore do not demonstrate the asymptotic O(1) requirements of Definition 2, and the abstract's phrase 'with a scalable thresholding scheme' could be read as an existence claim. The paper should state explicitly that the existence of scalable thresholding schemes is an empirical conjecture supported by small-system numerics, not a theorem proven here.

minor comments (4)

[End Matter title] The End Matter section title is misspelled as 'END MA TTER' (p. 6) and should be corrected to 'END MATTER'.
[Figure 2 caption] Figure 2's caption should specify whether the thresholding projector is built from the exact overlap matrix S or from the noisy matrix S + Delta S; this is essential for interpreting the numerical evidence in light of the proof gap described above.
[Equation (4)] The 'with high probability' statement after Eq. (4) is informal; a tail bound specifying the failure probability would help readers assess the claimed O(M/N_shots) scaling.
[Abstract] The abstract's phrase 'structured NOQE instances' is undefined; the numerical evidence covers only the two hydrogen families, so the scope of the empirical claim should be stated more precisely.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Corollary 2 is a conditional theorem whose O(M) shot bound is derived from the external Mathias-Li bound and a concentration inequality; the scalable-thresholding condition is an explicit input assumption, not the conclusion.

full rationale

The paper's central claim is a conditional theorem: Corollary 2 assumes Definition 2 (scalable thresholding) and derives N_shots = O(M/E^2) via the Mathias-Li bound (Theorem 1), Lemma 1's ||X||^2 ≤ kappa(S), and the random-matrix concentration chi = O(sqrt(M/N_shots)). The condition kappa(S_{>epsilon}) = O(1) is an input assumption, not the output; the output is a shot-count scaling that follows only after applying the external perturbation theorem and the concentration bound. No fitted constant is reused as a prediction: the numerical c values are varied and reported as evidence, and the paper explicitly labels the sublinear growth as 'suggest,' not as a derived result. The O(M^3) comparison cites [23] by the same group, but the identical bound is also attributed to [20] with independent authors, so the self-citation is not load-bearing. The unanalyzed issue that finite-shot thresholding uses eigenvectors of S + Delta S rather than exact S is a correctness/scope gap, not a circular reduction: Definition 1 and Corollary 2 explicitly define thresholding via the exact S, so the theorem does not secretly assume its own conclusion. Overall, the derivation chain is self-contained and no load-bearing step reduces to its inputs by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The core derivation uses a standard perturbation theorem, a Gaussian noise model, and a hypothesized scalable thresholding scheme. The only ad hoc parameters are the thresholding coefficients and the noise level in the numerical demonstration. No new physical entities are introduced.

free parameters (2)

Thresholding coefficient c in epsilon_N=cM(N) = c = 1e-3, 3e-3, 5e-3 (per family)
Hand-chosen constants used in the numerical demonstration to keep the thresholding bias visibly bounded over N=2..9. The theorem is conditional on a scheme with such properties, so the demonstration is evidence, not a proof.
Matrix-element noise standard deviation sigma = sigma = 1e-3 (Ha for H, dimensionless for S)
Fixed simulation parameter. The empirical claim of sublinear cost growth is inferred at this noise level rather than from a shot-count sweep, so it is a hand-set input to the numerics.

assumptions (5)

standard math Mathias-Li perturbation theorem (Ref. [18]) bounds eigenangle error via arcsin(chi||X||^2/d_min) under chi||X||^2 < min_j beta_j.
Unproved background theorem; the derivations of Corollaries 1 and 2 build on it directly.
domain assumption Finite-shot matrix-element errors are unbiased zero-mean Gaussians with variance sigma^2 = O(1/N_shots), giving chi = O(sqrt(M/N_shots)).
Used for Eq. (4); ignores correlations, Pauli-grouping prefactors, and non-Gaussian tails.
domain assumption Reference states are normalized so S has unit diagonal, giving lambda_max(S_>epsilon) >= 1.
Used in Corollary 2 to show d_min,>epsilon = Omega(1).
ad hoc to paper A scalable thresholding scheme exists (kappa(S_>epsilon)=O(1) and thresholding bias=O(1)) for the NOQE families considered.
Def. 2 qualifies the theorem; the paper gives numerical evidence for hydrogen chains and rings over N=2..9, not a general proof.
ad hoc to paper Thresholding is performed with eigenvectors of the exact overlap matrix S.
Def. 1 and the proof of Corollary 2 use V_>epsilon from exact S; a real protocol has only S+DeltaS, so this is an unproven practical premise.

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Cite this review

Pith. "Pith review of Improved Measurement Cost Scaling in the Nonorthogonal Quantum Eigensolver." pith.science (2026). https://pith.science/paper/U4XFZ4UU

@misc{pith2026260812830,
  author       = {Pith},
  title        = {Pith review of: Improved Measurement Cost Scaling in the Nonorthogonal Quantum Eigensolver},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U4XFZ4UU}},
  note         = {Machine review of arXiv:2608.12830}
}

abstract

Quantum subspace diagonalization methods are promising algorithms for quantum chemistry on near-term quantum computers. These methods can estimate low-lying energies of molecular systems using shallow quantum circuits, at the cost of many circuit repetitions to estimate the projected matrix elements. Errors in these matrix elements can be converted into much larger eigenvalue errors by an ill-conditioned overlap matrix. We study this bottleneck for the nonorthogonal quantum eigensolver (NOQE), which constructs a compact multireference subspace from dressed unrestricted Hartree-Fock states. We prove a finite-shot perturbation bound showing that, after overlap thresholding, the eigenvalue sensitivity is controlled by the condition number of the retained overlap matrix rather than by a worst-case dimension factor. With a scalable thresholding scheme, the upper bound on the per-matrix-element shot count required to reach a target accuracy scales as $\mathcal{O}(M)$, improving on the previously known $\mathcal{O}(M^3)$ bound, where $M$ is the number of reference states. Numerical experiments on hydrogen chains and rings suggest that, in practice, the measurement cost of structured NOQE instances can grow even more slowly than this linear bound.

Figures

Figures reproduced from arXiv: 2608.12830 by the authors.

**Figure 2.** FIG. 2. Performance of finite-shot NOQE with overlap thresholding for the ground state energies [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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